Of course it should be true "by translation of $c$; the polynomial is still a polynomial", but the actual algebra doesn't seem to be as easy.
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Mark HurdDec 8 '12 at 9:32

I'm not too clear on what the actual question here is. Are you asking that if the series is a polynomial for some value of $c$ then it is a polynomial for all values of $c$?
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EuYuDec 8 '12 at 9:35

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@EuYu: The question is: if $c\neq0$ is given and $\sum\limits_{n=0}^\infty {a_{n}}(x-c)^n$ is a polynomial, does it follow that for some $k$ one has $a_n=0$ for all $n>k$. The answer is yes.
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Marc van LeeuwenDec 8 '12 at 9:37

@EuYu What Marc said; I have updated the question.
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Mark HurdDec 8 '12 at 9:39

1 Answer
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The substitution $x:=x-a$ maps polynomials to polynomials, and since its inverse is the substitution $x:=x+a$, the result of the substitution into a series $S$ is a polynomial if and only if $S$ is itself a polynomial. Therefore $\sum_{n=0}^\infty {a_{n}}(x-c)^n$ is a polynomial if and only if $\sum_{n=0}^\infty {a_{n}}x^n$ is a polynomial.